4a2 b3 - 2a3 b3,

by simply factoring it.

second wire to denote the six tens, and so on,
moving into the space AF a sufficient number may be abbreviated to the expression,
of counters on each wire to denote the num-
2a2 b2 (2b — a),
ber of units of the corresponding order;
when the operation is complete, the given
number will be represented as in the diagram.
We have supposed the value of the unit, in
passing from wire to wire, to increase ac-
cording to the decimal scale; but the instru-hhd. for hogshead, lb. for pound, &c.
ment is equally applicable when the value of
the unit increases according to any scale,
either uniform or varying. If the duodeci-
mal scale be adopted, there will be required
eleven counters on each wire; if any varying
scale is used, the number of counters on any
wire must be at least equal to the number of
units of that order contained in a unit of the
next superior order, diminished by 1. The
method of recording a number constructed ac-
cording to any scale, is entirely similar to
that already explained.

An abbreviation is a single letter, or a simple combination of letters, standing for a

word or sentence: thus, A. stands for acre,

A-BRIDGE', [Fr. abréger, to shorten. Gr. ẞpaxvç, short]. To shorten, to contract. The abridgment of an expression in Algebra, is the operation of shortening it by substitution : thus, every equation of the second degree containing but one unknown quantity, is a particular case of the general form

ax2 + bx + c = 0;

or, dividing both members by a,

b

C

0;

a

a

During the middle ages, the abacus was which may be abridged by substituting 2p

used by bankers, money-changers, &c., but instead of a frame with wires and beads, they made use of a bench or bank covered with black cloth, divided into checks by white lines at right angles to each other. Counters placed upon the lower bar denoted pence, those on the second bar shillings, those on the third, pounds, and those on the fourth, fifth, sixth, &c., bars, denoted tens, hundreds, thousands, &c., of pounds.

ABACUS PYTHAGORICUS, or Pythagorean abacus. A table computed for the purpose of facilitating numerical calculations. It is nothing more than the multiplication table as given in ordinary treatises on arithmetic.

ABACUS LOGISTICUS, or sexagesimal canon. The same as the Pythagorean abacus or multiplication table, carried to 60, both ways.

AB-BRE'VI-ĀTE. [L. abbrerio, to shorten]. To reduce to a small compass, to epitomize.

AB-BRE-VI-A'TION, the operation of abbreviating or shortening. The abbreviation of a fraction is the operation of reducing it to lower terms; thus, if both numerator and denominator of the fraction be divided by 9, the fraction is said to be abbreviated, or reduced to In Algebra, an expression is said to be abbreviated when it is shortened by any algebraic process: thus, the expres

sion

b

C

for and 9 for giving the equation.

a

x2 + 2 px + q

= = 0.

The last equation is not only easier to remember, but is also under a simpler form for discussion.

The operation of abridging may generally be resorted to with advantage, whenever complicated expressions enter into long computations. After completing the computations, we can, if necessary, substitute for the symbols introduced, their values in terms of the original quantities employed.

AB-RUPT' POINT of a curve. A point at which a branch of a curve terminates: thus, the curve whose equation is y − b = (x − a) | (x − a) has an abrupt point for x = a. lar point.

See Singu

AB-SCIS'SA. [L. abscissus, ab, from, and scindo, to cut]. One of the elements of reference by means of which a point is referred to a system of rectilineal co-ordinate axes. If we draw two straight lines, in a plane, intersecting each other, one of them being horizontal, it has been agreed to call the horizontal one the axis of X, or the axis of abscissas, and the other one the axis of Y, or the axis of ordinates.

ABSOLUTE SPACE, is space considered without reference to material objects. or limits.

AB'STRACT. [L. abstractus, to draw from draw] Separate, distinct from something else. or separate; from abs, from, and traho, to

ABSTRACT EQUATION, is an equation expressing a relation between abstract quantities only, as,

3x+4x-5= 0.

In sucl: a system, the abscissa of a point magnitude represented by the equation passes Is the distance cut off from the axis of X by through the origin of co-ordinates. See Ana line drawn through it, and parallel to the alytical Geometry. axis of Y. All abscissas measured to the tight, are, by convention, regarded as positive, and consequently, all at the left must be conwdered negative. The abscissas of all points situated on the axis of Y, are 0. In space, :he term abscissa is applied in a more genral sense, and may mean a distance measured parallel to either of the horizontal axes, the distance measured on a parallel to the axis of Z being always called the ordinate. It is customary to define the abscissa of a point in space, to be the distance of the point from the co-ordinate plane YZ, measured on a line parallel to the axis of X. The rule for signs is analogous to that employed in a plane system; all distances measured towards the right are considered positive, those to the leit must be negative; the abscissas of points m the plane YZ are 0. When the term abscissa is appiied to distances measured from une plane XZ and parallel to the axis of Y, they are considered positive when measured m front of the plane, and negative when measured behind it. When the int lies in the plane XZ, the abscissa is 0.

AB'SO-LUTE, [L. absolutus, av, from, and solvo, to loose or release]. Complete in itself, independent. The absolute term of an equation, is that term which is known, which does not contain the unknown quan thus, in the equation

ax2 + bx2 + cx + d = 0,

d is the absolute term. If we regard
term as involving the unknown quantity ni
some form, then the absolute term is that in
which the exponent of this quantity is 0. In
every entire equation, the absolute term is
equal to the continued product of all the roots
of the equation with their signs changed.
Hence, if the absolute term of an equation
is 0. one or more of the roots of the equation
must be equal to 0.

ABSTRACT QUANTITY, is one which does not involve the idea of matter, but simply that of a mental conception; it is expressed by a letter, symbol, or figures: thus, the number three represents an abstract idea, that is, one which has no connection with material things whilst three feet, presents to the mind an idea of a physical unit of measure, called a foot. So a "portion of space bounded by a surface, every point of which is equally distant from a point within, called the centre," is a mere conception of form. When we call it a sphere, we employ the term to express our idea of the abstract magnitude.

All numbers are abstract when the unit is

abstract. Arithmetic, which treats of the relations and properties of such numbers, is ab

stract arithmetic. This embraces the whole

science and theory of arithmetic: Concrete or Denominate Arithmetic being nothing more than the art of applying the principles developed in Abstract Arithmetic to Denominate Numbers.

Since Algebra differs little from ordinary A.timetic, except in the nature of the lanuge employed, we must regard the Science otlgeura as purely abstract. In Geometry also, in magnitudes considered, viz., lines surfaces and solids, are mere mental conceptions of extent and form, which are represented by geometrical figures. The discussion of these magnitudes in the development of their relations and properties is, therefore, In Analytical Geometry, the equations em- necessarily confined to abstract quantities. ployed are indeterminate; that is, they involve We may, therefore, 1egard Geometry as an more unknown quantities than there are equa- Abstract Science. And generally, all the tions, and the absolute term is that one which principles of Mathematical Science are deis independent of all the unknown quantities veloped from a consideration of abstract quan or variables. It may be demonstrated that tities only.

when the absolute term is 0, the geometrical What is usually termed Abstract Mathe

which is called declivity. Acclivity implies ascent, declivity implies descent.

matics, or pure mathematics, involves the tra-distinction to that of descending ground, entire science, whilst that which is called Concrete, or Mixed Mathematics, is nothing more than the art of applying previously developed principles to physical objects, as suggested by the demands of society.

A'CRE [L. ager, land. Gr. aypoç, a field] A unit of measure employed in land surveying. In the United States, the standard acre contains 4840 square yards, or 43,560 square feet. In the form of a square, one side would measure about 69.5701 yards, or 208.7103 feet. The acre contains 160 square rods, or perches. The subdivisions of the acre are roods and perches. The acre containing 4 roods, and the rood 40 perches.

There are 640 acres in a square mile, hence, an acre is theth part of a square mile. The English statute acre is the same as that of the United States.

The Irish acre contains 1 acre 2 roods

121

AB-SURD'. [L. absurdus, ab, from, and surdus, deaf, insensible]. A proposition is absurd, when it is opposed to a known truth. The term absurd, is used in connection with a kind of demonstration called the "reductio ad 'absurdum." In this kind of demonstration, a certain proposition is assumed as true, and is combined with known truths by a course of logical arguments, thus deducing a chain of conclusions until one is arrived at which disagrees with a known truth, when the original supposition or hypothesis is pronounced absurd, and its contrary is considered proved. 19,27 perches English. As an example of this mode of demonstration, we may instance the proposition to show that, "If two straight lines have two points in common, they will coincide throughout." In this proposition it follows that if they do not coincide, they must enclose a space which is manifestly impossible; hence, the proposition that they do not coincide involves an absurdity, and the proposition is said to be reduced to an absurdity. This method of proof, though sometimes objected to as unsatisfactory, is, nevertheless, as strictly logical, and as conclusive as any other method. The reasoning is quite as perfect, and the conclusions equally irresistible.

A-BUND'ANT NUM'BER. A number which is less than the sum of all its aliquot parts: thus, 12 is an abundant number, be

cause

12 < 1 + 2 + 3 + 4 + 6. An abundant number is distinguished from a perfect number, which is equal to the sum of its aliquot parts, and from a deficient number, which is greater than the sum of its aliquot parts.

AC-CI-DENT'AL POINT of a line. In Perspective, the point in which a line drawn through the point of sight, and parallel to the given line, pierces the perspective plane. It is a point of the indefinite perspective of the line. See Vanishing Point.

The Scotch acre contains 1 acre 1 rood 321 37 perches English.

The Welsh acre contains about 2 English

acres.

The Strasbourg acre is about one-half of an English acre.

tains 4,088 square yards, or nearly of an The French acre or arpent of Paris conEnglish acre.

The French woodland arpent contains 6,108 square yards, or about 1 acre 1 rood 1 perch English.

In the new decimal system of France, the Are contains 119.603 square yards, the Decare 1196 03 square yards, and the Hecatare 11960.3 square yards.

A-CUTE', [L. acutus, sharp pointed]. Sharp as opposed to obtuse. An acute angle, is one that is less than a right angle. In degrees, an acute angle is less than 90°.

ACUTE-ANGLED TRIANGLE, is one that has all of its angles acute.

ACUTE CONE, is one in which the vertical angle of the meridian triangle is less than 90°, or less than a right angle.

ACUTE HYPERBOLA, is one whose asymptotes make an acute angle with each other. In it, the transverse axis is always greater than the conjugate.

AC-CLIV'I-TY, [L. acclivus, from ad, to, ADD, [L. addo, from ad, to, and do. to and clivus, an ascent]. In Topography, the give]. To unite or put together, so as to steepness or slope of rising ground, in con- form an aggregate of several particulars.

AD-DI'TION, [L. additio, from addo, to 3. There is a third method of proof, which give to]. The operation of finding the sim- is only applicable to numbers written in the plest equivalent expression for the aggregate decimal scale. It is called the method by of two or more quantities of the same kind. casting out the 9's. The principle on which Such expression is called the sum of the this method depends requires some eluciquantities dation. Since

In arithmetic, the quantities to be added are always numbers, written either according to the decimal scale, or according to some vary ing scale. In the first case, the operation of adding is called Addition of Simple Numbers, in the second, Addition of Denominate Numbers. The operation in both cases are identical in principle, and may be described as follows: Write down the numbers to be added so that units of the same order or denomination shall fall in the same column.

Add together the units of the lowest order, and divide their sum by the number of such units contained in one of the next higher order: set down the remainder, and carry the quotient to the next column. Continue the operation till the column of units of the highest order is reached, and set down the entire sum of that column

ADDITION OF DECIMALS The rule in this case does not differ from that already given. In fact, every number written in the scale of tens is a decimal, whose value depends upon the place of the decimal point. When this point is fixed, the orders are counted from it in both directions. When numbered to the left, they are called orders of entire units; when to the right, they are called orders of decimals.

ADDITION OF VULGAR FRACTIONS. Reduce all the fractions to equivalent ones having a common fractional unit. Add the numerators

together, and write their sum over the denominator of the fractional unit: the result will be the sum required.

PROOF.-There are several methods to verify the accuracy of the operation of addition.

1. If the columns of units have been added from the bottom upwards, let them be added from the top downwards; the results should be the same.

10 = 9+1, 100 991, 1000 = 999 + 1 and so on, it follows that if a number expressed by 1 followed by any number of 0. be divided by 9, the remainder will be 1. Again,

202(91). 200

2000 =

2(99+1),

2 (999+1), &c.; hence. if a number expressed by 2 followed by any number of O's be divided by 9, the remainder will be 2. Generally, if a number expressed by 3, 4, 5, 6, &c., followed by any number of O's, be divided by 9, the remainder will be 3, 4, 5, 6, &c.

It is evident that if we divide each of the parts by 9, and then divide the sum of the remainders found, by 9, the final remainder will be the same as that which is found after dividing the entire number by 9. Any number, as 5634, may be written

5000+ 600 +30 +4.

and from the preceding principle it tollows that if any number be divided by 9, the remainder will be the same as that obtained by dividing the sum of its digits by 9

Upon these principles is based the following rule:

Take the sum of the digits in each number to be added, and having divided each sum by 9. set down the remainder in a column at the right.

Take the sum of these remainders and divide it by 9. setting the remainder beneath. If this remainder is the same as that found by diriding the sum of the digits in the sum total by 9. the work is probably correct.

EXAMPLE.

Excess of 9's. 4567 4

3214 1

1187 8

8968 | 4

2. Separate the numbers to be added into two or more groups, and add these groups, each by itself, and take the sum of the results; The sum of the digits in the first number In the this sum should be the same as that first is 22, and the remainder found, 4. obtained. The principle on which this method second number, the sum of the digits, 10, and depends is, that a whole is equal to the sum of the remainder 1. In the third, the sum of all its parts. the digits is 17, and the remainder, 8. The

sum of these remainders is 13, and the remainder 4, which is also the remainder is an adfected equation, containing terms obtained by dividing 31, the sum of the digits which involve different powers of 1 See in the sum total, by 9. Hence, we conclude Affected. that the operation of addition was correctly performed.

ADD'I-TIVE. A quantity is additive AD-JUST'MENT. [From ad, to, and just when it is preceded by a positive sign. If it us, just.] The operation of bringing all the is not preceded by any sign, the sign is parts of a mathematical instrument into thei always understood. proper relative positions. When the part: have these positions, the instrument is sai to be in adjustment, and is fit for use. When several independent steps have to be taken